## Exploring the Polynomial (x+2)²(x-1)²

The expression (x+2)²(x-1)² represents a polynomial function. Let's delve into its characteristics and how we can analyze it.

### Understanding the Structure

**Factors:**The expression is formed by two factors: (x+2)² and (x-1)². Each of these factors is a square of a binomial.**Expansion:**We can expand the expression to get a polynomial in standard form: (x+2)²(x-1)² = (x² + 4x + 4)(x² - 2x + 1) = x⁴ + 2x³ - 3x² - 4x + 4**Degree:**The highest power of x in the expanded form is 4, making it a fourth-degree polynomial.

### Key Properties

**Roots:**The roots of the polynomial are the values of x that make the expression equal to zero. Since the expression is factored, we can easily find the roots:- (x+2)² = 0 implies x = -2 (multiplicity 2)
- (x-1)² = 0 implies x = 1 (multiplicity 2)

**Symmetry:**The graph of the polynomial will be symmetrical about the line x = -1/2. This is due to the even powers of both factors.**End Behavior:**As x approaches positive or negative infinity, the polynomial will approach positive infinity. This is because the leading term is x⁴, which has a positive coefficient and an even power.

### Graphing the Polynomial

To visualize the polynomial, we can use the following steps:

**Find the x-intercepts:**Plot the roots at x = -2 and x = 1.**Determine the y-intercept:**Set x = 0 and evaluate the polynomial: y = 4.**Analyze the end behavior:**The graph will rise on both ends.**Consider the multiplicity of the roots:**The multiplicity of 2 for both roots indicates that the graph will touch the x-axis at these points but not cross it.

By combining these steps, we can sketch a rough graph of the polynomial.

### Applications

This type of polynomial function can arise in various applications, including:

**Modeling physical phenomena:**The equation could represent the motion of an object or the shape of a curve.**Solving engineering problems:**It could be used in calculations related to structures, circuits, or other engineering systems.**Financial modeling:**The polynomial could model growth patterns or investment returns.

By understanding the properties and behavior of (x+2)²(x-1)², we gain valuable insights into its applications and its role in various fields.